Small cancellation groups with and without sigma-compact Morse boundary

TL;DR

This paper constructs examples of small-cancellation groups with both sigma-compact and non-sigma-compact Morse boundaries, revealing how this property distinguishes their quasi-isometry types and providing a complete characterization for certain groups.

Contribution

It presents the first known examples of groups with non-sigma-compact Morse boundary and characterizes when $C'(1/6)$ groups have sigma-compact Morse boundary.

Findings

Some small-cancellation groups have non-sigma-compact Morse boundary.

The property of sigma-compactness distinguishes quasi-isometry types.

Complete characterization of sigma-compact Morse boundary in $C'(1/6)$ groups.

Abstract

We provide examples of classical small-cancellation groups which have non-sigma-compact Morse boundary. These are first known examples of groups with non-sigma-compact Morse boundary. Some small-cancellation groups do have sigma-compact Morse boundary, so this property distinguishes quasi-isometry types of small-cancellation groups. In fact, we give a complete description of when Morse boundaries of $C'(1/6)$--groups have sigma-compact Morse boundary. We also provide examples of $C'(1/6)$--groups where all Morse rays are strongly contracting.